Serial binary arithmetic units



L.S.HECHT SERIAL BINARY ARITHMETIC UNITS Feb. 3, 1959 5 Sheets-Sheet 1 Filed April l. 1954 INVENToR. fm1-' f. Me'cw L.. s. HECHT 2,872,111

SERIAL BINARY ARITHMETIC UNITS 1954 5 Sheets-Sheet 2 Feb. 3, 1959 L. s. HEcHT SERIAL BINARY ARITHMETIC UNITS 3 Sheets-Sheet 5 Filed April l. 1954 K .WNN

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United State Patent O SERIAL BINARY ARITHMETIC UNITS Lester S. Hecht, Los Angeles, Calif., assignor to Hughes Aircraft Company, Culver City, Calif., a corporation of Delaware This invention relates to serial binary arithmetic units, and more particularly to serial binary adders, subtracters, and adder-subtracters wherein the result digit in each binary place is formed as a function of the associated input signals and the result digit and input signals of the preceding binary place, no carry function being required.

Many adders, subtracters, and adder-subtracters are presently known, wherein the vbinary carry series and result, either sum or difference, are formed through sepa rate networks, except for the inclusion of a delayed carry signal in the sum or difference producing network. Such circuits, for example, are described in U. S. Patent No. 2,609,143, entitled Electric Computer for Addition and Subtraction, by G. R. Stibitz, issued September 2, 1952, and U. S. Patent No. 2,643,820, entitled Circuit for Adding Binary Numbers, by F. C. Williams et al., issued June 30, 1953. In the patents by G. R; Stibitz and F. C. Williams et al., the binary carry series is generated by a separate logical network and applied to a one-binarydigit delay circuit, which may be a flip-flop. Each binary sum digit is then produced by a separate result-producing network as a function of the corresponding input signals and the preceding carry signal stored in the delay circuit.

One disadvantage of the prior-art binary adders 'and subtracters is that a carry flip-flop is required for each such unit used in the computer. This requires the addition, to the computer, of carry hip-flops if they are not available from other arithmetic circuits within the computer.

A second disadvantage of the prior-art binary adders, subtracters, and adder-subtracters occurs when the result signal series is utilized to control other circuits, such as a decimal correction circuit. Result signals of greater power than that produced by a logical network are then required, necessitating a buffer output circuit and associated circuits for coupling the logical network to the output circuit.

Even where an output buffer is not needed for power requirements, complementary result signals may be required. ln this case, a complementer circuit is needed, a suitable complementer circuit being described in copending U. S. patent application, Serial No. 308,045, now Patent No. 2,812,451, for Complementary Signal Generating Networks, by Daniel L. Curtis, filed September 5, 1952.

The above and other disadvantages of the prior art are obviated, according to the present invention, by forming each binary result digit as a function of the binary input signals in the corresponding binary place and the input signals and result signals in the preceding binary place. Each binary result digit signal is entered into a storage element as it is formed, and the'storage element then provides an output signal indicating the result signals in the preceding binary place as the next result signal is formed. In this manner, the carry signals are effectively replaced by the input signals and the result signals in the corresponding binary place. The storage element may,

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in a particular application, also be utilized as a buffer circuit or complementer, thus obviating the necessity of an additional buffer stage and complementer circuit.

-As will become apparent from the ensuing description, the techniques provided by the present invention are readily applicable for use in any numbering system employlng binary digit signals representing magnitudes or quantities in accordance with a selected code, examples of such systems being conventional binary, binary-coded decimal, and binary-coded octal numbering systems. In order to facilitate aclear and concise description of the present invention, however, it is assumed at the outset that the serial binary arithmetic unit of the present invention is adapted to operate in a conventional binary numbering system. A conventional binary numbering system may be defined as a numbering system wherein a magnitude or quantity is represented by a group of binary digits having weights or powers of 2. Accordingly, each binary digit of a group has a weight double that of the immediately lesser order binary digit and one-half that of the immediately greater order binary digit of the group. It is further assumed that each binary input number to the arithmetic unit of the present invention is represented by series of binary electrical signals received in the order of least significant binary digit signal first and most significant binary digit last. Similarly, corresponding binary digit signals of the result are serially formed in the same order. kThus a binary digit signal of a signal series has a weight or significance twice that of a binaryudigit signal in the preceding binary place of the same series.

In its basic structural form the invention comprises a logical gating means, which receives a first set of input signals representing binary digits of the numbers and a second set of input signals representing binary digits in the preceding place of the numbers, for producing output signals, one output signal being produced foreach binary digit of the result number. A storage element receives the output signals and produces a series of result-repro.

senting signals, each binarydigit of the Vresult number being represented by a signal signal.

For convenience, the logical gating means may be considered as divided into a first and second gating circuit; the first gating circuit receiving the first set of input signals and producing a series of first output signals having first` and second levels indicating equality and inequality, respectively, between the input signals in the corresponding place; and a second gating circuit responsive to the first output signals and the second set of input signals for producing a second set of output signals which are fed to the storage means.

A single or two-input storage element may be used as the storage means. Where the storage element has only a single input circuit, the second gating circuit is mechanized to set the storage element directly to 1 or 0 according to the corresponding result digit. In this situation, the result signals, representing the delayed result digits, must actually be applied to the second gating circuit. However, where the storage element is a bistable element, such as a flip-flop, having separate 1 and 0 input circuits,a simplification may be effected in the second gating circuit by defining separate 1 and 0 setting functions. In this case, it is not necessary to actually apply the result signals to the second gating circuit, since the second output signals produced by this gating circuit are defined on the basis of the memory characteristic of the storage element. This definition recognizes the fact that if the storage element is already in the desired state, no

signal need be applied to its l-setting input circuit andv signal need be applied to its* that if it is in the O state, no 0-setting input circuit.

' In a modified embodiment of the present invention, the

function of the first gating circuit may be performed inaf comparator circuitwhich is already available in the computer. A comparator circuit suitable for this operationrisl operated to provide the series off-first outpi'ltsig'rialseindi-'- eating equality or inequalityV between corresponding binary input digits v Accordingly, it is an object of the'inveh'tionto `p`1'o1ici`e`-' a serial binary arithmetic unitl foraddition orsubtrc'tioin' requiring no carry iiip-op, the result being formedv as va`E function of the input signals and previous result and input signals.

Another object is to provide a serial binaryari'thrhe'tic unit wherein the result signals are-,directly entered into an output storage element, thusv eliminating the need of av separate butter output'stage or complementer circuit.

A further object is to provide-a serial binary adder, subtracter, oradder-subtracter wherein' the result signals are entered `into an output flip-flop having separate l and input circuits andl wherein the result signals are formed as a function of the corresponding input signals and thepreceding input signals Yet another object is to: provide a serial binary adder, subtracter, or adder-subtracter producing complementaryV output signals, requiring only a single bistable element.

The novel features which are believedl to be characteristicof'the invention, both as .to its organization and themeth'od of operation, together with further objects and advantagesfthereof, will be better understood vfrom the following description consideredl in connection with the `accmpanyingdrawings in which several embodiments of the inventionare illustrated byway ofexamples. It isY to be expressly understood; however, that the drawings are for the'purpose of illustration and'descr'iptionl only and are not intended as a'deiinitio'n of the limits of the invention. v

Fig. 1 is a' block diagram of aserial binary arithmetic unit according to the present invention;

Fig. .2 is a schematic diagram of a serial binary adder according to the present invention;

Fig. 3 is a schematic diagram of a serial binary subtracter according to the present invention;

Fig. 4 is a schematic diagram of a serial binary adder.- subtracter of the present invention; and

Fig. 5 is an alternative species of the serial binary adder-subtracter of Fig. 4, wherein a separate serial comparator circuit is utilized.

Referring now to the drawings, there is shown in Fig. l an arithmetic unit 100 for" performing an operation of addition or subtraction upon tirst and second numbers'A and B, for producing the corresponding binary result R. The numbers A and B are represented by electrical signals Aj and Bj, respectively; and the immediately preceding binary digits of the numbers A and'B are represented by the electrical signals A 1 and B,' 1, respectively. Each binary digit of the corresponding binary result is represented by an electrical signal Rj. The subscript j is utilized to indicate a binary-digitposition or an equivalent time interval, and the subscript j--l is utilized to indicate the next lower-order binary-digit position or a binary digit signal delayed one binary digital time interval. As has been previously discussed; each ofthe numbers A and B and the corresponding result number R is represented by a series of binary electrical signals arriving in the order of least significant binary signal first and most signiticant4 binary signal last. Accordingly, a binary digit signal delayed one binary digit place-and represented by a symbol accompanied by a subscript j-l has a weight or signiticance one-half that of a presently received corresponding signal represented by the same symbol` but accompanied by the subscript j. l

As shown in Fig. l, a logical gating'rneansV 10G receives the input signals Ai,.Bj-,rAj-n1, `andBj 1, andpr duce an output signal series which may be defined in several manners;` as'will be'explained below. A delay'ot one binary position to obtain the signals A14 from the signals A! and to obtain the signals B, 1 from the signals Bj may be obtained by known techniques such as through an electrostatic delay line or through a mercury delay line. The delay may also be provided'. by introducing the signals Aj to aiirst flip-dop to produce the signals A744 and by introducing the signals B3 to a second p-o'p' to producethe signals B141.

The logical` gating. meansltlG'may be divided'. into. a iirst gating circuit 10Q and a second gating circuit 10R for convenience of description. The first gating circuit 10Q receivesthe' input signals A! and B, and produces complementary signal 'Qj and Q3, representing equality and inequality, respectively, between the corresponding A5 and B, signals. The second gating circuit ltlR is responsive to the complementary signalseries Qj. and Q1 and the signal'se'rie's A14. and`B',- 1,.folr producing a src-r ond outputsignal series which` is applied to ,theT storage; element 10S.

Before. describing thedetailed structu re a1:1d` operation@ of the serial binary adder-subtracter of Fig. 1, it is ad-l vantageous to consider the basic concepts involved in binary addition and subtraction. One of the most common methods of developingthese. concepts. is to form tirst a.` truth table involvingl all the possible combinations ofvariables under consideration, and from the truth table-- Yderive Vfunctions representing the inter-relationships ofthese variables. The sum function. for the addition OEA- and` B maybe derived from the following-truth table:

represent the binary sum and carry digits, and where respectively;

'b -l represents the previous carry digit. y From the table above, the tunctionfor the Sum plus (-i) the logical inclusive or.

From Table I it is also noted that the carry,

C? may be Written as a function of the sum R? and the' binary input digits AJ and B3 as follows:

6 'Them by substitution ofthe value of c, I C+ Referring now to Fig. 2, there is shown a serial binary l" adder mechanized according to the above principles of ifi the Sum eXPression above, the latter CXPICSSOII bef binary addition. As shown in Fig. 2, the storage element comes: 5 Rf=tagliata.,+En.@H+BaniitAi-Bf-Fn+ [Raul-+EnJp.ifafaitati.) which may be written in another form as: l Rear-uitgaans.taf-Balla)+,.1.E. 1. Af.+a2.n,n+

Rental-Battatami-n+@gaaf-licitan1 Letting QJ be a Signal which iS in a 1-1'Cp1eSemil1g 10S of Fig. 1 is a iiip-op Rs having 1 and 0 input cir- State when A1=Ba and 1n a 0-represenf1ng State when 15 cuits and producing complementary output signals Ai; the expression Q, may be written as follows: R d

i-i an i-i (Q) 1 representing the delayed binary sum of input numbers Q (A E B) A and B. Flip-dop Rs is controlled through a second 1' j 1" i 20 gating circuit 20Rs producing signal series And substituting in the expression for 1R, and 0R, l I R; which are applied to the 1 and 0 input circuits, respecabovwe obtain' Y. tively, of the flip-opfRs. Signal series Ri:Ri-i-l(i-ilBi-i)-Qi'l-Ai-i-Bi-i-Qil'i' 1R; and 0R;

In a similar manner, the subtraction functions for the i-i ffOIl arSSigDal S0U1Ce,1PUtSgDa1S By-i aIId '1-i, subtraction of B from A toformthe difference. from a second signal source, and complementary signals R31 Q3 and QJ, produced by a first gating circuit 20Q which may be derived from the following Table H: :si aFsigecilfic mechanization of the first gating circuit 10Q -Table II Before considering the specific mechanization of logi-v cal gating circuit 20RsV of Fig. 2, it is necessary to con- A, B, 0,11 C- vRd; VQ-

sider the general form of equations defining the-input l i functions for flip-Hops.: The discussion here is brief, since.. o o 0 o o 1 the theory of flip-op control functions is discussed in o o 1 1 i 1 considerable` detail in copending U. S. patent applicag i g l g tions seriai No. 327,567,110w Patent No. 2,816,223, for4 1 o o o i o 46 Binary-Coded Flip-Flop Counters, by Eldred C. Nelson, (l) g 3 (l) filed December 23, 1952, and Serial No. 327,131 for i i 1 i i i BinaryfCoded Flip-Flop Counters, by Robert Royce Johnson, filed December 20, 1952. From Table II, the expression for the carry, Flip-flop Rs in Fig. 2 is assumed to be a conventional flip-flop having 1 and 0 input circuits, such that pulses may be written as, C applied separately to the 1 and 0 input circuits set the ip-op to l and 0 states, respectively. For example, 01.-: R ('i-} 15,.)..R';! Bl the ip-iiop may be constructed in a manner similar and to that described on pages 164 to 166, inclusive, of vol- Ci= R Ai Bi+-g1 (l4i+ Bf) 50 ume 19Jentit1ed Waveformsfof the Radiation Laboratory Series, published by the Massachusetts Institute of and the prevlous carry C, Technology in 1949. It should be understood, however, as follows, l that other types of flip-flops may beutilized, provided v that they are controlled through a properly mechanized Cli=Ri.(AfeilBi-i)+Ri.A'ii.Bf-1` 55 logical gating circuit. For example, `an overriding .i d ip-op may be utilized with the Boolean equations de- C"`I R`I`A' B 1+Rf"(Ari-Bf) lining the logical gating circuit for indicating when the R: C, 1.(A,.B,+A,.Bf)+ C' 1.(A,-.B,+ ALB) ip-op is to be set to 1, regardless of its previous state.

tiliziri7 an overriding ip-op pulses are continu- Again substituting Q, for the expressions showing In u P AjzBj and Q1 for the expressions indicating AB the 60 ously applied to the 0 input circuit of the flip-flop, so that it is set to 0 unless the logical gating circuit conditionsv functlon for d are satisfied, in which case a pulse is applied to the 1 b ,tt R1 input circuit of the flip-op overriding the pulse applied may e Wfl en to the 0 input circuit and setting the flip-flop to l.

R=CT1Q1+ -iQi Three general types of flip-op input functions may be and substituting for utilized to control the sequence of stable states of an Cl;l associated flip-op. According to one type of equation, the difference function becomes: the lsequence of stable states of the ip-flop is directly defined so that the value of the equation (1 or 0) at aV d d [Rf" A""B'-1+R"(A"I+BY`I)]'Q' tion. When a setting function is utilized, the ip-flop which may be written in another form as: must be an overriding flip-flop of the type just described,

i the gating output signal into complementary signals. R ,.[(A,..1+B,-1).Qi+ A;-,.B, 1.Q,] 75 According toak second type ofdeiining equation,A the two partial-changing functions, separately dening the v conditions for changing the associated dip-nop stable; state from Oto l and from l to 0. The partial-changing functions are particularly useful where the equations include the output signals of the ip-op to be controlled. In this case, the partial-changing,functions may be simpliiied according to rules briefly considered' below which are fully described in the above-mentioned copendingapplications to. Nelson and Johnsoin The settingf changing, and simplified partial-changing functions for controlling a flip-flop Fk (k representing any flip-hop) are designated, respectively, by Vthe notations: toFk=; lFk=OFk=g and 1Fk=, and OFk=, respectively.

As is more fullyv explainedl inthe above-mentioned copending applications to Nelson and Johnson, any flipop function may be written inthe form:

toFk=F'k.G{-7F7.

Fl and k representing the complementary output signals offlip-op Fk. l This. may be reduced to the. simplified partial-changing functions:

the form of simplied partial-changing functions as follows:

mechanizing functions for the gating circuits ZGQ and 20Rs inv Fig; 2' as follows:

Q1-=,(5-Bji-1-j)c2 v @Fmt-Baalen@ andY 1R.=[(Aif1i Bi.-i)Qi'lAf-iBfxQilP (,ZtlRs) RilAi-li-i-ri- ('.'i-r-lrBi-zlQl-CP where the signal` cp is introduced to indicate a synchroj nizing and condition, signal cp occurringv once each binary-digit time interval.

Each yof the and functions in the above equations is provided in Fig. 2 with an and circuit which responds to signals applied to separate input terminals and producingV l-representing output signals only when all input signals are l-representing signals. Thus, and circuit ZlQ-l in the gating circuit ZOQ responds to signals Aj and Bj and produces a l-representing output signal when bgthAi and Bj are l-representing signals. Similarly, and`circuits ZGQ-ZV, ZllQ-S, and ZOQ4 respondto separatelyjv angliqtigtlt. sigtlaflslto1 produce. the, l-repxresenting.

signal dened by the corresponding and" functions in the equation (20(2).

Eachof the or\functions in the above equation is provided by an orcircuit responding to separately applied input signals and producing a l-representing output signal when anyone or-more of the input signals isin a Vl-r'presenting state. ',Thus, or circuit ZllQ-S responds -.to signals generated by the and circuit 20Q-1 and the and circuit ZtlQ-Zfwhich are applied to the separate input terminals, and produce arl-representing Qi signal whenone or both of'theseinputs isa l-representing signal. In asimilar manner, each and and or function of equation (2011s) is provided with an and and or circuit, respectively, in the second gatingA circuit 20Rs.

The operation of the binary-adder section of the binary adder-subtracter1 of' the Present invention presented in" Fig. 2 may be illustrat'edrfrom` Table III below which depicts a binary time-sequence function of the output. signals of the gating circuits 20Q and 20Rs, and the As indicated inTahle Ill', the output series each digit time interval during which b oth Aj and Bj are the same value, that is both in l-representing states or both in G-representing states; while Qj is in the O-representing state when signals A, and B1 are dissimilar in" value.

The one input,

to the Rs flip-flop will be eiective when either A, 1 or B54 is inthe l-representing state, and the signal Q] is also in the l-representing state; or when both A 1 and B, 1 are in the O-representingstate, and signal Q3 is also in the O-representing'state.

Referring, nowV to Fig. 3, a serial binary subtractor according to the present invention is presented. A first gating circuit 30Q generates complementary output signals QJ and QJ. It is noted that the gating circuit 3tlQ is similar to the iirst gating circuit 20Q of Fig. 2, in that the signal Q, is in al l-representing, state when input signais A, and Bj are both in the l-representing state or are i both in the O-representing state; and the signal Q3 is in a l-representing state when Aj and Bj are in opposite representing states. A second gatingcircuit 30Rd produces a pair of control signals as a function of the complementary signals Q, and .QL

control signals, respectively, produces complementary output signals Rg., and RL1 representing the dilerence of input numbers A and B, delayed one binary-digit time interval.

The second gating circuit SllRd is mechanized in conformity with the functions for binary subtraction previously derived. on flip-flop theory, these functions may be written in the form of mechanizing setting function as follows:

which may also be written in the form of simplified partialchanging functions as:

The second gating circuit 30Rd of Fig. 3 is mechanized in accordance with the above equations (30Rd); and in light of the previous explanations for mechanization of logical functions, further detailed consideration is deemed unnecessary.

The operation of the binary subtractor of Fig. 3 is illustrated in Table IV below, wherein the input Bj is subtracted from the input A,.(A,--Bi) of the binary numbers used in Table III.

Again it is noted that the nal difference series is generated after a delay of one binary-digit time interval.

A combined binary adder-subtractor of the present invention is illustrated in Fig. 4. The circuit presented in Fig. 4 may be considered as a combination of the serial binary adder presented in Fig. 2 and the serial binary subtracter presented in Fig. 3, with the sign signals S and respectively indicating that the complementary signals R,- 1 and fia- 1 represent the difference and sum. The signals R1 1 and t- 1 are produced by a ipop R having l and 0 input circuits responsive to control signals lR] and ORJ, respectively, produced by a second gating circuit 40K as a function of input signals A,- 1, 1, B, 1, and l 3,- 1, and the complementary signals Qj and Qj. The signals Q1 and Q5 are produced by a rst gating circuit 40Q which is similar to the rst gating circuits ZllQ and 30Q of Figs. 2 and 3 respectively.

By introducing the sign signals S and into the mechanization functions (ZORs) and (SllRd) for addition and subtraction, respectively, these functions may be combined into a single set of mechanization functions, thus:

In the light of' the previous discussion l the binary input signals used in the previous tables are utilized.

TableV 0R, (Adcn Ri-i OR, (Subtract) S 0 0 0 0 1 1 0 0 Rf-x In Fig. 5 an alternate design of the adder-subtracter of the present invention is presented. As noted in the ligure, the circuit is similar to that depicted in Fig. 4, except for the inclusion of a comparator circuit in the first gating circuit, 50Q.

The signals Q3 and Q, may be obtained by slight modilication of the comparator already available in a digital computer. Such a comparator may be of the type described in copending U. S.' patent application, Serial No. 394,411 for Electronic Magnitude Comparator, by Robert Royce Johnson, led November 25, 1953. This comparator compares the equality or inequality of two compared groups of four binary digits. In the comparator disclosed in the above-mentioned copending application, a ip-op Q is initially set to the l-representing state by an input (ToCom) generated at the start of each series of four binary digit pairs to be tested. If any one pair of incoming binary digits is composed of unequal binary. digits, the Q flip-Hop is triggered to the O-representing state and remains in this state for the remainder of the group being tested. At the beginning of the next series COCCO group to be compared, the flip-flop is again set to the 1- representing state by the signal (ToCom).' This comparator is illustrated in Fig. 5, wherein the signal input (ToCom) is received on line 50Q-3 and the inequality signal is generated on line 50Q-4.

The partial-chauging functions for the Q llip-op may be expressed as follows:

VIl

It is noted from these :partial-changinglfunctions that the Q Hip-nop is, setto the. l-representing state after the comparison commandsignal (ToCom); and is set to the (l-representing'statel after any'binary-digit period during which AHl and BH1 are unequal; but is notagain returned to 4the. 1-representingi state.. until the v (T 0Com) signal is agai'nreceived..

In order toV make the comparator useable for generatingthe equality land" inequality signals. Q! andi-Q3 after each pair of binary digits is compared, an equality function.(X4-Y).(AHbBiH--inlpl) is logically added to the l-input functionof flip-flop' Q j Signals X and; Y are subtraction and-.addition commands, respectively. The partial-changing functions fonip-op Q then become:

The signal (ToCom-)- is.. applied to` one input of the or gate 50Q-5 by way of Aline 5,0Q6, Vand the output of the or gate SHQ-5 is applied. through an and gate 50Q-2- to'the l input terminal of the Q flip-dop. Thesignal. (ToCom) which was formerly connected to the input line 50Q-3 of the comparator is disconnected, asindicatedby the X in the drawing.

As is well known in the prior art, the complementary output signals of a liip-iiop` are` delayed one clock time interval, i. e., the ip-op changes state in response to the input signals at the fallof the clock. timing pulse.. Since the signals QJ and Q3, generated by the second gating circuits in Figs. 2, 3'; and 4, are generatedwvithout this digit time-interval. delay,l the input signals Am and BH1, preceding the inputisignals Aj and Bj by one1digit time interval, must. be used in the comparison circuit of Fig. 5. As the input signals A541 and BH1 are readily available from the magnetic-drum circulating register, the utilizationL of these signals involvesy no additional circuitry.,Y

The second gating circuit SGR and the ip-op R are similar to that described in relation to Fig'. 4; and, therefore, further detailed explanation of the. remainder of Fig. 5 vis deemed unnecessary. The first and second gating circuits of Fig. 5 are mechanized in accordance with the following mechanization equations:

been pointed out that the previous input signals are readily available' from the computer components, such as an in- (SGR) put circulating register, andtherefore no additional cir-- cuitry is required to obtain these signals.

' It has been further shown that by placing the result in a single bistable storage element, the previous result signals are readily available because of the inherent delay characteristics of sucha bistable storage element.

the result signals become available in the form Ofcomplementary signals of sufficient power to drive other logical circuitry of the computer, thus eliminating the need of'an additional buffer stage or complementer circuit..

AV further simplication of the adder-subtracter of the present invention was achieved' by. entering the result signal directly.' intoaf. nip-dop. havingT separate 1 and 0 It was; also explained that by using a bistable. storage element,.

l2 inputicircuits; Because of the'characteristics of such a tiip-op, it was shown that the result signals may be; directly entered into the nip-flop as a function of the present and previous input signals, the previousresult signals being effectively accounted for Within thetlip-op. Y While-the embodiments herein havedescribedserial binary, adders, subtracters, and adder-subtracters, it should be understood that the principles taught arereadily applicable to. parallel adder-subtracters and sub-combinations thereof. It should also ybe understood that the logical mechanization equations which were utilized for defining the logical gating. circuityare genericto an entire classof equations which'may be derived therefrom by well known algebraic manipulations.

It should befurther understoodl that the diodes used in the gating circuits may be replaced by any class-Ofcomponents performing the same or similar function such as electron emitting tubes, transistors, selenium .rectitiers, or the like. It should also b'e understood'that the electronic dip-flops employed may be replaced by any storage or delay element such as a magnetic, dielectric, or sonic delay line.

Although thepresent invention has been described in" relation to binary electricalpulses, it should be apparent that the principles herein taught are equally applicable to anytwo-condition signal system such as marky spacecsymmetrical wave, or. carrier modulation system. It should: be'` further apparent that. the. present invention is not limited to purely binary number systems but is. equally operable with any binary digit coded system such as the binary-coded decimal system, the binary-coded octal system, and thelike.

The embodiments herein described utilize electrical signals,electrical diodes, and -electronic ip-ops, but it 'shouldbe clearly understood that the principles herein taught are equally applicable to electro-mechanical, mechanical, hydraulic, or`chemical components having unirectional features, two stable states, and storage capacity.

What is claimed as new is:

1. An arithmetic unit for performing an operation of addition or'subtraction upon first and second numbers to produce the corresponding result, each number being represented by a diii'erent pair of input signal series, each of said pairsincluding aseiies of rst input signals and a series of'second nputsignals, each rst input signal representing a binary digit in a corresponding binary-digit place of the associated number and each second input signal representing a binary digit in the preceding next lower-order binary-digit place of the associated number, the result including a series of binary digitsrepresented by a corresponding series ofresult signals; said arithmetic unit comprising: tirst means, responsive to the first input signals` for producing a series ofrst output signals, each rst output signal. indicating equality between the corre-- sponding rst input signals; second means coupled. to said trst .means and responsive to. said first output signals and the second input signals for producing a series of second output signals; and storage means coupled to said second means and responsive to `said second ouput signals for producing a series-of output signals representing the corresponding result.

2. An arithmetic unit for performing an operation of addition .or subtraction upon first and second numbers; to produce the corresponding result, each number being represented by a different pair of-input signal series, each of said pairs including a series of rst inputsignals` and a series of second input signals, each iirst input signal representing a binary digit in the corresponding place of the associated number and each second input signaLrepresenting the binary digit in the preceding next lowerorder binary-digit place of' said number, said result including a series of binary digits represented by a correspending seriesof. result signals; said arithmetic unit comprising; signal combining means, responsive to each pair vof tirst input signals, for producing a two-level signal having rst and second levels representing equality and inequality, respectively, between the corresponding pair of first input signals; output circuit means coupled to said signal combining means and responsive to said two-level signals and the second input signals, for producing a series 5 of control signals; and storage means coupled to said output circuit means for receiving said control signals and producing output signals representing thebinary digits of the result.

3. The arithmetic unit defined in claim 1, wherein said storage means is a single input bistable element, directly set to its bistable states by said second output signals and producing complem-entary output signals representing -said result.

4. The arithmetic unit defined in claim 1, wherein said second output signals consist of first and second control signals and wherein the storage means is a flip-flop having 1 and 0 input circuits coupled to said second output means and responsive to said rst and second control signals, respectively, for producing complementary output signals representing said result.

5. An arithmetic unit for performing an operation of addition or subtraction upon first and second numbers A and B, to produce the corresponding result, each binary digit of said iirst number being represented by an electrical signal designated A,- and the immediately preceding next lower-order binary digit by an electrical signal designated A7 1, and each binary digit of said second number being represented by an electrical signal designated Bj and the immediately preceding next lower-order binary digit by an electrical signal designated B14, and each digit of said result num-ber being represented by an electrical signal designated R5; said adder-subtracter comprising: first means, responsive to electrical signals A, and B3 for producing a series of first output signals, each signal of said first output signals indicating similar digit values in signal Aj and the corresponding signal Bj; second means coupled to said iirst means and responsive to said first output signals and signals A,- 1 and B, 1, for producing a second output signal series, one second output signal being produced upon occurrence of each signal A44; and a bistable storage element coupled to said second means and responsive lto said second output signals for producing a third output signal series corresponding to the result delayed one binary-digit time interval.

6. The arithmetic unit defined in claim 5, wherein said storage means is a single input ip-iiop directly set to its stable states by said second output signals and producing complementary output signals complementary signals :Ri-1 and i-x representing the sum 14 where-the dot and the plus represent the logical and and or respectively, and the bar represents the complementary state and the signal R; y representing a setting relationship for triggering the flipop to a particular state and for maintainnig the ip-op in that state.

7. The arithmetic unit defined in claim 5, wherein said second output signals consist of a first and second control signal series designated respectively, and wherein said storage means is a flip-flop coupled to said second means and having a 1 and a 0 input circuit responsive to said first and second control signal series, respectively, and producing complementary output signals designated the dot and the plus (l-) representing the logical and" and or respectively, and the bar the complementary state and indicating the introduction of a signal to the 1 input circuit of said flip-flop and ORg indicating the introduction of a signal to the 0 input circuit of said flip-flop.

8. The adder-subtracter defined in claim' 5, wherein the binary result represents the difference of B from A, where:

in said second output signals directly represent a difierence series and wherein said bistable storage element is a single-input flip-flop directly set to its bistable states by said second output signals and producing complementary output signals representing the delayed result; and wherein said output means are connected to receive the input signals Aj and B3 and the immediately preceding input signals A, 1 and B 1 and to operate in a particular interrelationship on said signals for the production of signals in accordance with the following equation:

A, 1 and B,- 1 and to operate in a particular interrelationship on said signals for the production of signals the dot and plus (-1) representing the logical and v or, respectively. and the bar over the signals the avenir a lf nd :Rf-1 representing the difference: of B from A'; and wherein'th'e output circuit means are connected to receive the input signals. Aj and Bj and the immediately preceding input signalsfAjLj-andiBjgj and to-,operate on said signals in .a

particularinterrelationship for the production of output signals in accordance with .the following equations:

the dot and the plus (4I) representing the logical and and or,, respectively, andthe bar (-),I over they signals the complementary state and 1R;i indicatingv the introduction of a signal to the 1 input circuit of said flip-flop and v indicating the introduction. of asignal tothe (l. input circuitof said flipflop.

10. The serial binary arithmetic unit defined in-claim 5, wherein said first output signals consist of a pair of complementary electrical signal series designated Qj and Qj, respectively; and. wherein saidiirst outputz: signal. means are connected to receivefthe'input.signals-Ajand Bj and to operate on saidsignals in a particular interrelationship for the production of signals in accordance with the, following equations:

the dot' and the plus` (l) representing the logical and and on respectively, and the bar over thev signals the complementary state.

l1. An arithmetic unit for adding first and second numbers A and B, each binary digit of numbers A and' B being represented by b'inaryelectrical signals Aj and Bj, respectively, andi the immediately precedingnext lowerorder binary digits of numbers A and B by binary electrical signals Aj j and Bj j, respectively; said unit comprising: a first gating circuit, including a' plurality'4 of logical and and or circuits, for producingcomplementary signals Qj and Qj, representing equality'and inequality, respectively, between corresponding Aj andV Bj signals; a second gatingcircuit, including a plurality ofY logical and and or" circuits,.saidrsecond gating circuit being coupled to said first gating circuit and responsive to signals Aj j, B, 1, Qj, and @j for prodncinga-pair oticontrol signals 1R; and OR;

respectively; and aip-fl'op Rs having separatel and 0- input circuits responsive to said signals lR and 0R;

respectively,` and producing complementary electrical' 12. YAn 'arithmetic unit for subtracting a second num: ber B v from a first number A, each digit ofnumbers A and B' being represented by binary electrical signals Aj and'Bj',l respectively, .and the immediately preceding next lower-order. digits of numbers A and B by binary electrical signals A,- 1 and B,- 1,y respectively; said unit com,- prising;A a rst" gating circuit including a plurality of logical and and or circuits, for producing comple-` mentary4 electrical signals Qj and Qj representingv equality andl inequality, respectively, between corresponding Aj and Bj signals; a second gating circuit, including a plurality vof 'logical and and or circuits, coupled to said first gating circuit and responsive to signals Ajh B4. Qi, and Qi', vfor producing a pair of control signalseries 1R;l and ORE1 and ay flip-flop Rd, having separate 1 and 0. input circuits coupledto said second gating circuit and responsive to saidcontrol signal series respectively, andA producing complementary electrical signals' Rit-1 and i1-1 representing the difference of B from A.

13; Anarithmetic unit for performing addition or subtraction in response to electrical sign signals andS, respectively; upon numbers A and B and producing the corresponding resultA number R, each digit of numbers A and'B being represented by binary electrical signals Aj. and'Bj, respectively,land the immediately preceding next lowerorder digits of numbers A and B by binaryA electrical*v signals Aj 1 and,B,- 1, respectively; said unit comprising: a tirst'gating circuit, including a plurality of logical and and or circuits, for producing complementary signals Qj and Qj, representing equality and inequality, respectively, between corresponding Aj and Bj, signals; a second gating circuit, including a plurality of logical and and or circuits, responsive to signals` Aj 1, B`,- 1,V Qj, and Q5, and to the signvsignals and S` for producing a pair of control signals lRj and ORj, respectively to represent the results of binary addition or subtraction in accordance with the electrical sign signals S and S, respectively; and a flipdiopR, having separate 1 and 0 input circuits, responsive to said lRj and ORj signals, respectively, and producing complementary electrical signals Rj, j and tj1 representing the result-nurn ber R.

14. An arithmetic unit .for performing an operation of adidtion or subtraction upon first and second numbers A and B in response to complementary sign signals S and S to produce the corresponding result, each binary digit of said first number being represented by an electrical signal deginated Aj and the immediately preceding next lower-order binary digit by an electrical signal designated A, 1, each binary digit of said second number being represented by an electrical signal designated Bj and the immediately preceding next lower order binary digit by an electrical' signal designated Bj- 1, and each binary digit of said result number being represented by a pair of complementary electrical signals designated Rj and j; said= adder-subtracter comprising first means, responsive toelectrical signals Aj and Bj and connected to produce complementary pairs of first output signals Qj and Qj in accordance with the logical equations:

QFAi-Birl-Air where the dot and the plus (f-l) represent the-logical and and on respectively, and the bar overA a symbolv of a Ysignal indicates the complement of the signal; second meanslcoupled` tosaid first means and responsive to said iirst output signals Qj and Qj and signals Aj 1 and BH and connected to produce a second output signal series consisting of a pair of control signals designated lRj and ORJ, respectively, in accordance with the logical equations: 1R1= [Bi-i- S +11 +5 -Ai11 Qri- [41(S+4 1)+S11]l 0R1= J1(S+A1-1)+AJ1]Q1+ [B 1 1- (S 'F1-1) +S111 -i where the parentheses and brackets represent the logical and; and a bistable ip-op coupled to said second means and having one and zero input circuits,

18 responsive to said 1R, and 0R, signals, respectively, and producing complementary output signals vRJ and R, representing the binary result.

References Cited in the file of` this patent UNITED STATES PATENTS 2,571,680 Carbrey Oct. 16, 1951 2,775,402 Weiss Dec. 25, 1956 FOREIGN PATENTS 1,041,729 France June 3, 1953 1,044,678 France I une 24, 1953 

